Polynomial Time Code For 3-SAT Released, P==NP

An anonymous reader writes "Vladimir Romanov has released what he claims is a polynomial-time algorithm for solving 3-SAT. Because 3-SAT is NP-complete, this would imply that P==NP. While there's still good reason to be skeptical that this is, in fact, true, he's made source code available and appears decidedly more serious than most of the people attempting to prove that P==NP or P!=NP. Even though this is probably wrong, just based on the sheer number of prior failures, it seems more likely to lead to new discoveries than most. Note that there are already algorithms to solve 3-SAT, including one that runs in time (4/3)^n and succeeds with high probability. Incidentally, this wouldn't necessarily imply that encryption is worthless: it may still be too slow to be practical."

Comments

Probably Wrong but Clearly Falsifiable

[eldavojohn]

Even though this is probably wrong, just based on the sheer number of prior failures ...

Okay, so I'm going to agree with you that it's probably wrong. After reading the paper once last night in a sort of sleepy state, it's certainly a novel way of massaging each 3-Sat problem into an expanded space of memory (compact triplets structures) and then reducing this to solve the problem (all within polynomial time).

So the great thing about this paper is that it's short, it's not theoretical (which is a double-edged sword)/has been implemented (in Java, no less) and it's incredibly falsifiable. I'm not intelligent enough to poke holes in his proofs for this work but people can apply the code to DIMACS formatted problems to their heart's content and if they find one that produces the wrong answer then this is falsified.

I hope we find someone capable of commenting on the proof involving the transformation of the problem from compact triplets formula to compact triplets structure and the hyperstructures presented in the paper. If this is legit, the 3-Sat problem is one that more complex problems are reduced to in order to show that said complex problems are NP-complete. And that's something that's been proved by the Cook-Levin theorem and given in the classic Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnson.

Refreshingly tangible implementation, if I may say so myself!

Re:Probably Wrong but Clearly Falsifiable

[dvdkhlng]

Maybe I'm overlooking something, but to me it looks like they're doing the reduction to a polynomial-time problem already at the very beginning of the paper (I guess if there is a fault, there it hides). As soon as they go to "compact triplet" structure, the instance of 3-SAT is polynomial-time solvable using a trellis algorithm. Yes, very similar to the algorithm that is employed to decode convolutional codes.

In fact they're decomposing the initial 3-SAT problem into multiple "compact triplet" 3-SAT problems intersected using an AND operation. But as these intersected 3-SAT formulas use the same variables, without any interleaving (permutation) applied, the trellis algorithm still applies (just like solving for a convolutional encoder with > 1 check bits per bit).

Thinking once more about that: the compact triplet structure is clearly not general enough to express generic 3-SAT problems. This is like attempting to transform a quadratic optimization problem x^T*H*x involving a symmetric matrix H into a corresponding problem with a tri-diagonal matrix H.

The only way I see they could do the transform is by introducing exponentially many helper variables, thus moving the problem back into NP again. But it does not look like they're attempting something like that.